---
title: "Piezoelectric load cell and charge amplifier"
subject: "Instrumentation and Measurement"
chapter: 7
key: "semester_4/instrumentation_and_measurement:piezoelectric-load-cell-charge-numerical"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/piezoelectric-load-cell-charge-numerical
site: BCT Engineering
---

# Piezoelectric load cell and charge amplifier

Instrumentation and Measurement · Chapter 7: Common Mechanical Measurement System and Transducers

- Asked in: year not given, 4+4 marks
- Video: [Charge amplifier selection explained in 4 minutes](https://www.youtube.com/watch?v=ow1H9bJllug) — Kistler Group
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/piezoelectric-load-cell-charge-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-7

## Question

(a) Explain the principle of a piezoelectric load cell. Why is a charge amplifier used, and why can it not measure static force?

(b) A quartz crystal of charge sensitivity 2.3 pC/N is loaded by 500 N. The total capacitance of crystal and cable is 1.2 nF. Find the open-circuit voltage. If a charge amplifier with a 100 pF feedback capacitor is used, find its output. Also find the lower cut-off frequency if the leakage resistance is 1 G$\Omega$.

## Short answer

Quartz/PZT develops charge q=dF under force; a charge amplifier with feedback capacitor makes output q/Cf independent of cable capacitance. Static force cannot be held because charge leaks through insulation/input resistance. For 500 N: q=1.15 nC, open-circuit voltage 0.958 V, amplifier output 11.5 V, lower cutoff 0.133 Hz.

## Answer

#### (a) Piezoelectric load cell
Certain crystals and ceramics (quartz, Rochelle salt, barium titanate, lead zirconate titanate (PZT)) develop an electric charge on their faces when a force deforms them (the direct piezoelectric effect):

$$
q = d\,F
$$

where $d$ is the charge sensitivity (C/N). The crystal is a self-generating (active) transducer, mounted between two plates in a stiff housing. It is equivalent to a charge source $q$ in parallel with a capacitance $C$, so the open-circuit voltage is $V = q/C$.

```text
        F
        v
   [=========]  <- top plate
   | crystal |  charge +q / -q on faces
   [=========]
   housing -> output
```

**Why a charge amplifier:** the voltage $q/C$ depends on the cable capacitance, so changing the cable changes the reading. A charge amplifier (an op-amp with a feedback capacitor $C_f$) gives $V_o = -q/C_f$, which is independent of cable length, and it has a high input impedance.

**Why not static:** the charge leaks away through the insulation and the amplifier input resistance $R$. With a time constant $\tau = RC$, the output decays under a constant force, so only dynamic (and quasi-static) forces can be measured. Advantages of the cell: very high stiffness, high natural frequency, wide dynamic range and small size.

#### (b) Numerical
Charge generated:

$$
q = dF = 2.3\times10^{-12} \times 500 = 1.15\times10^{-9}\ \text{C} = 1.15\ \text{nC}
$$

Open-circuit voltage ($C = 1.2$ nF):

$$
V = \frac{q}{C} = \frac{1.15\times10^{-9}}{1.2\times10^{-9}} = 0.958\ \text{V}
$$

Charge amplifier output ($C_f = 100$ pF), in magnitude:

$$
|V_o| = \frac{q}{C_f} = \frac{1.15\times10^{-9}}{100\times10^{-12}} = 11.5\ \text{V}
$$

(The sign is negative for an inverting charge amplifier.)

Lower cut-off frequency ($R = 1$ G$\Omega$, $C = 1.2$ nF):

$$
f_c = \frac{1}{2\pi RC} = \frac{1}{2\pi(10^{9})(1.2\times10^{-9})} = 0.133\ \text{Hz}
$$

**Answer: $q = 1.15$ nC; $V = 0.958$ V; charge-amplifier output $= 11.5$ V (magnitude); $f_c = 0.133$ Hz.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does a piezoelectric load cell produce when force is applied?
   A. A change in resistance
   B. A change in inductance
   C. An electric charge q = dF on the crystal faces
   D. A change in capacitance only
   Answer: C. The direct piezoelectric effect generates charge proportional to force, making the crystal a self-generating transducer. (Key point: Piezoelectric principle)

2. Why is a charge amplifier used?
   A. Its output q/Cf is independent of the cable capacitance
   B. It increases cable capacitance
   C. It makes the cell measure static force
   D. It removes the need for a crystal
   Answer: A. Open-circuit voltage q/C depends on cable capacitance, while Vo = -q/Cf does not, and the input impedance is high. (Key point: Charge amplifier)

3. Why can a piezoelectric cell not measure a constant static force?
   A. The crystal stops generating charge instantly
   B. The charge leaks away through insulation and input resistance with time constant RC, so the output decays
   C. The crystal becomes magnetic
   D. The amplifier saturates immediately
   Answer: B. Leakage makes the output decay for a constant force, so only dynamic or quasi-static forces are measured. (Key point: Static force)

4. For 2.3 pC/N and 500 N with a 100 pF feedback capacitor, what is the charge amplifier output?
   A. 0.958 V
   B. 1.15 V
   C. 11.5 V
   D. 115 V
   Answer: C. q = 1.15 nC, and Vo = q/Cf = 1.15e-9/100e-12 = 11.5 V. (Key point: Numerical)

## Related in Instrumentation and Measurement

- [Full-bridge cantilever load cell output](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Acantilever-full-bridge-output-numerical) — Instrumentation and Measurement, chapter 6
- [Ballast circuit for a strain gauge](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Aballast-circuit-strain-gauge-sensitivity) — Instrumentation and Measurement, chapter 6
- [Elastic transducers and strain gauge load cells](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Aelastic-force-transducers-load-cells) — Instrumentation and Measurement, chapter 7
- [Hydraulic and pneumatic load cells](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Ahydraulic-pneumatic-load-cells) — Instrumentation and Measurement, chapter 7

## Related in other subjects

- [Piezoelectric transducer sensitivity and output voltage](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Apiezoelectric-transducer-output-voltage) — Instrumentation I, chapter 3
- [Capacitance and charge matrices (9.81 m clearance)](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Acapacitance-charge-matrix-9-81m) — High Voltage Engineering, chapter 5, asked 3×
- [Capacitance and charge matrices, transposed and untransposed](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Acapacitance-charge-matrix-midspan) — High Voltage Engineering, chapter 5
- [Coaxial cable charge density, D and E](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acoax-cable-charge-density-d-e) — Electromagnetism, chapter 2